几类非线性分数阶微分方程边值问题解的存在性

【作者】 李红霞；

【导师】 杜新生；

【作者基本信息】 曲阜师范大学 ， 应用数学， 2017， 硕士

【摘要】 应用拓扑度方法,锥上的不动点理论、半序方法,本文研究了几类非线性分数阶微分方程边值问题解的存在性.全文共四章,第一章介绍有关非线性泛函分析、分数阶微分方程的基本定义、理论以及重要的不动点定理.第二章在文献[22] [23]的基础上,研究如下非线性分数阶微分方程边值问题利用Banach压缩映像原理得到其正解的唯一性结论,利用Leray- Schauder二择一定理,得到了解的存在性,同时利用Leggett - Williams定理得到了解的多重性结果.第三章在文献[24][25]的基础上,利用锥拉伸压缩不动点定理研究如下边值问题得到了带变号非线性项的分数阶微分方程奇异边值问题正解的存在性结果,第四章在文献[31][32] [33][34]的基础上,我们研究如下带有积分边界条件的分数阶微分方程利用不动点指数,Banach压缩映像原理,并借助算子的第一特征值和第一特征函数,得到其正解的唯一性结论.更多还原

【Abstract】 By using the method of topological degree, the fixed point theory on cone, and the partially ordered method, the existence of solutions to several class of nonlinear fractional differential equations boundary value problems was considered in this thesis.The thesis is divided into four chapters.In the first chapter, the basic definitions and theories and some impor-tant fixed point theories of nonlinear functional analysis and fractional differential equations was introduced.In the second chapter, on the basis of [22] [23], the following boundary value problem for nonlinear fractional differential equation[D0α+u(t) + f(t, u(t)) = 0, t ∈ (0,1), 1 < α ≤ 2(4)u(0)=0,u’(1)=0.was considered. By using the Barnach contraction mapping principle, we obtain the uniqueness of its positive solution, the existence of solution was considered by using Leray - Schauder alternative theorem, at the same time, by using theLeggett- Williams theorem, we obtain the multiplicity results.In the third chapter, on the basis of [24] [25], by using the Guo -Krasnoselskill’s fixed point theory, the following problem Dαx(t)+f(t,x(t-τ)) = 0,t∈(0, 1)\{τ},x(t) =η(t), t ∈ [-τ,0], (5)x’(0) = x"(0)=x"(1)=0.was discussed. We get the existence of positive solution to a singular boundary value problem for fractional differential equation with chang-ing sign nonlinearity.In the fourth chapter, on the basis of [31] [32] [33] [34], we study the following fractional differential equation with integral boundary condi-tion D0α +u(t)f(t,u(t))=0,0<t<1,(6)u(0) = u(1) = 0, u(1) = ∫01g(s)u(s)ds.The uniqueness of positive solution was considered by using the fixed point theory, Barnach contraction mapping principle, the first eigenvalue and the first characteristic function of the operator.更多还原